Acceptance sampling is a form of product control in which a random sample is drawn from a lot, inspected, and a decision is made to accept or reject the entire lot based on the sample results. It is a compromise between no inspection and 100% inspection.
| Aspect | 100% Inspection | Sampling Inspection |
|---|---|---|
| Cost | Very high — every item examined | Much lower — only a fraction inspected |
| Time | Slow; not feasible for large lots or continuous production | Quick; lots released faster |
| Destructive testing | Impossible (all items destroyed) | Feasible (only sample destroyed) |
| Inspector fatigue | High — leads to missed defects (≈ 80–90% effective) | Less fatigue — more careful inspection |
| Risk of wrong decision | Nearly zero (if no fatigue errors) | Non-zero — some bad lots accepted, some good lots rejected |
| Information gained | Only accept/reject | Also provides quality history and trend data |
A manufacturer produces 50 000 light bulbs per day. Testing each bulb's lifespan by burning it to failure is destructive and would destroy the entire output. Sampling inspection (e.g., testing 200 bulbs) is the only feasible option.
A supplier ships 5 000 capacitors. Inspecting all 5 000 at ₹2 per capacitor costs ₹10 000. A single sampling plan (n = 200, c = 3) costs only ₹400 per lot and still provides 95% confidence of detecting a lot with more than 3% defectives.
Producer's risk (\(\alpha\)): The probability of rejecting a lot that actually meets the quality standard. This hurts the producer — a good lot is wrongly returned.
Consumer's risk (\(\beta\)): The probability of accepting a lot that does not meet the quality standard. This hurts the consumer — a bad lot is wrongly accepted.
Conventionally, \(\alpha\) is set at 0.05 (5%) and \(\beta\) at 0.10 (10%). These are not symmetric — the consumer's risk is typically set lower because accepting bad quality is usually more harmful than rejecting good quality.
A lot with \(p = 0.01\) (1% defective — a "good" lot) is submitted. The sampling plan has \(\alpha = 0.05\). This means there is a 5% chance the lot will be rejected even though it meets the quality standard. The producer bears the cost of the false rejection.
A lot with \(p = 0.06\) (6% defective — a "bad" lot) is submitted. The sampling plan has \(\beta = 0.10\). This means there is a 10% chance this bad lot will be accepted. The consumer receives a substandard batch.
The Operating Characteristic (OC) curve is a plot of the probability of accepting the lot (\(P_a\)) against the lot fraction defective (\(p\)). It completely characterises the discriminatory power of a sampling plan.
Using the Binomial model, for a lot with fraction defective \(p\):
Using the Poisson approximation (when \(n\) is large and \(p\) is small, with \(\lambda = np\)):
Plan: \(n = 50\), \(c = 2\). Compute \(P_a\) for \(p = 0.02\):
\(P_a = P(X \le 2) = \binom{50}{0}(0.02)^0(0.98)^{50} + \binom{50}{1}(0.02)^1(0.98)^{49} + \binom{50}{2}(0.02)^2(0.98)^{48}\).
\(= 0.3642 + 0.3716 + 0.1858 = 0.9216\).
For \(p = 0.08\): \(P_a \approx 0.226\) — the plan discriminates well between 2% and 8% defective lots (compare the Poisson value 0.238 in Example 2).
Same plan: \(n = 50\), \(c = 2\). For \(p = 0.02\), \(\lambda = 1.0\).
\(P_a = e^{-1}(1 + 1 + 0.5) = 0.3679 \times 2.5 = 0.9197\). Very close to the Binomial value 0.9216.
For \(p = 0.08\), \(\lambda = 4.0\): \(P_a = e^{-4}(1 + 4 + 8) = 0.0183 \times 13 = 0.238\).
Acceptable Quality Level (AQL): The maximum fraction defective that is considered satisfactory as a process average. It is the quality level at which the producer's risk is \(\alpha\). Lots at AQL should be accepted with high probability (\(P_a = 1 - \alpha\)).
Lot Tolerance Percent Defective (LTPD): The worst tolerable quality level — the fraction defective at which the consumer's risk is \(\beta\). Lots at LTPD should be accepted with only low probability (\(P_a = \beta\)).
| Parameter | Stands for | Associated risk | Typical value |
|---|---|---|---|
| AQL | Acceptable Quality Level | Producer's risk \(\alpha\) (usually 0.05) | e.g., \(p = 0.01\) |
| LTPD | Lot Tolerance Percent Defective | Consumer's risk \(\beta\) (usually 0.10) | e.g., \(p = 0.05\) |
A sampling plan is designed so that AQL = 1% with \(\alpha = 0.05\) and LTPD = 5% with \(\beta = 0.10\). This means: lots with 1% defective are accepted 95% of the time; lots with 5% defective are accepted only 10% of the time. The OC curve must pass through the points (0.01, 0.95) and (0.05, 0.10).
If AQL is tightened from 1% to 0.5% (same \(n, c\)), the producer's risk increases — more good lots get rejected. If LTPD is lowered from 5% to 3% (same \(n, c\)), the consumer's risk increases — more bad lots get accepted. To maintain both \(\alpha\) and \(\beta\) at desired levels, a larger sample size is needed.
Average Outgoing Quality (AOQ) is the average quality of the product after the inspection process — i.e., after rejected lots have been 100% inspected and all defectives replaced with good items. It is a function of the incoming quality \(p\):
where \(N\) is the lot size. For large \(N\) relative to \(n\), this simplifies to:
\[ \text{AOQ}(p) \approx P_a \cdot p. \]Average Outgoing Quality Limit (AOQL): The maximum value of AOQ over all possible values of incoming quality \(p\). It represents the worst-case average quality the consumer will receive, regardless of the supplier's quality level.
Plan: \(n = 100\), \(c = 3\), \(N = 5000\). For \(p = 0.02\), \(P_a = 0.857\) (from Poisson, \(\lambda = 2\)).
AOQ = 0.857 × 0.02 × (5000 − 100)/5000 = 0.857 × 0.02 × 0.98 = 0.0168 ≈ 1.68%.
The average outgoing quality when incoming quality is 2% is about 1.68%.
For the plan \(n = 100\), \(c = 3\), compute AOQ for several \(p\) values:
| \(p\) | \(P_a\) | AOQ ≈ \(P_a \cdot p\) |
|---|---|---|
| 0.01 | 0.981 | 0.0098 |
| 0.02 | 0.857 | 0.0171 |
| 0.03 | 0.647 | 0.0194 |
| 0.04 | 0.433 | 0.0173 |
| 0.05 | 0.265 | 0.0133 |
| 0.06 | 0.151 | 0.0091 |
Maximum AOQ ≈ 0.0194 at \(p ≈ 0.03\). So AOQL ≈ 1.94%.
Average Sample Number (ASN): The average number of items inspected per lot under the sampling plan. For a single sampling plan, ASN = \(n\) (always inspect exactly \(n\) items).
Average Total Inspection (ATI): The average total number of items inspected per lot considering both the sample inspection and (if the lot is rejected) the 100% screening of the remainder:
When the lot is accepted (\(P_a\) close to 1), ATI ≈ \(n\). When the lot is rejected (\(P_a\) close to 0), ATI ≈ \(N\) (full inspection).
Plan: \(n = 80\), \(c = 2\), \(N = 2000\). For \(p = 0.02\), \(P_a = 0.783\) (Poisson, \(\lambda = 1.6\)).
ATI = 80 + (1 − 0.783)(2000 − 80) = 80 + 0.217 × 1920 = 80 + 416.6 = 496.6 items.
On average, about 497 items are inspected per lot when incoming quality is 2%.
For the same plan, at two different quality levels:
When the supplier's quality is good, most lots are accepted and ATI is close to \(n\). When quality is poor, most lots are rejected and ATI approaches \(N\) — the screening cost becomes enormous.